CN113077373A

CN113077373A - Image encryption method based on chaos mapping and bidirectional operation Feistel structure

Info

Publication number: CN113077373A
Application number: CN202110306051.7A
Authority: CN
Inventors: 佟晓筠; 刘西林; 张淼; 王翥
Original assignee: Harbin Institute of Technology Weihai
Current assignee: Harbin Institute of Technology Weihai
Priority date: 2021-03-23
Filing date: 2021-03-23
Publication date: 2021-07-06

Abstract

The invention provides an image encryption method based on chaotic mapping and a bidirectional operation Feistel structure, which is used for solving the problems that the conventional image encryption algorithm based on chaotic mapping is insufficient in scrambling-diffusion and is separated from a plaintext in an encryption process and easy to attack by the plaintext. Firstly, using SHA-256 algorithm to generate 256-bit hash value for a plaintext image, and calculating and correcting the hash value to obtain an initial value of a chaotic system; then updating the initial value of the iterative chaotic system, generating a pseudo-random sequence by the iterative chaotic system again, and calculating and correcting the pseudo-random sequence to obtain a round key and a modulation matrix; secondly, performing two-time bit-level scrambling, diffusion and pixel rearrangement operations on rows and columns of the image by using a bidirectional operation Feistel structure; and finally, carrying out global scrambling and diffusion operation on the sequence by using a modulation matrix to generate a final ciphertext image. Experimental results and security analysis show that the encryption method provided by the invention enhances the cleartext sensitivity of the algorithm, can effectively resist attack and has good security.

Description

Image encryption method based on chaos mapping and bidirectional operation Feistel structure

Technical Field

The invention belongs to the technical field of digital image encryption, and particularly relates to an image encryption method based on chaotic mapping and a bidirectional operation Feistel structure.

Background

With the rapid development of information technologies such as the internet and the like, multimedia becomes a main way for spreading information, and images as carriers of information have the characteristics of vivid image and large amount of information, and become an indispensable part of communication in daily life of people. In real life, military secret-related images and private images of individuals or businesses are often leaked and tampered, so that the security of image data is important to protect, and the image encryption technology is a method for effectively protecting and transmitting digital image information.

Because the image has the characteristics of large data volume, high redundancy, strong pixel correlation and the like, the traditional encryption algorithm is not applicable any more. The characteristics of the chaotic system such as initial value sensitivity, pseudo-randomness, non-periodicity and the like are consistent with the characteristics required by cryptography. At present, the chaotic encryption technology is widely applied to the field of information security, in particular to the field of image encryption^[1-7]。

The Feistel structure is a block cipher structure proposed by cryptologist Horst Feistel, is a symmetric cipher algorithm capable of being rapidly realized, and has great development potential in the technical fields of information encryption, hiding, authentication and the like^[8]。

Disclosure of Invention

Aiming at the problems of insufficient scrambling-diffusion, large time consumption, separation from a plaintext in an encryption process and easy attack to the plaintext in the existing partial image encryption method, the hyperchaotic Lorenz system is combined with a Feistel structure to construct a bidirectional bit-level scrambling, diffusion and pixel rearrangement for a plaintext image, and then global scrambling and diffusion operation are carried out, so that the aim of encrypting a transmission image is fulfilled.

The invention relates to two main modules, wherein the first module is bit-level scrambling, diffusion and pixel rearrangement of an image Feistel structure; the second module is global scrambling and diffusion of the image.

1. Bit-level scrambling, diffusion and pixel rearrangement of image Feistel structure

1.1 Generation of round keys and modulation matrices

By using SHA-256 calculations on the plaintext image informationThe method generates a hash value of 256 bits, and calculates the hash value to generate an initial value x of the chaotic system₀，y₀，z₀And w₀。

The invention adopts a hyper-chaotic Lorenz system, which is a famous four-dimensional chaotic system, the system has complex dynamic behavior, and the equation is as follows:

wherein x, y, z and w are state variables of the system, a, b, c and r are control parameters of the system, and the system is in a hyperchaotic state when a is 10, b is 8/3, c is 28, r is not less than-1.52 and not more than-0.06.

Using SHA-256 algorithm for different plaintext images will generate different 256bit hash values, which are expressed in decimal notation as k₁,k₂,…,k₆₄Divide them into 4 groups, i.e. { k }₁,k₂,…,k₁₆}，{k₁₇,k₁₈,…,k₃₂}，{k₃₃,k₃₄,…,k₄₈And k₄₉,k₅₀,…,k₆₄Calculating an initial value x of the hyperchaotic Lorenz system according to the following formula₀，y₀，z₀And w₀Given the system parameter values, i.e., a is 10, b is 8/3, c is 28, r is-1, after iterating the hyper-chaotic Lorenz system 2000 times using the initial values, the values of x, y, z and w are generated as the updated initial values of the system;

continuously iterating the chaotic system M multiplied by N times by using the updated initial value to generate 4 pseudo-random sequences x (a)₁,a₂,…,a_M×N)，y(b₁,b₂,…,b_M×N)，z(c₁,c₂,…,c_M×N) And w (w)₁,w₂,…,w_M×N) Generating 4 pseudo-random matrices by computing corrections to the 4 pseudo-random sequencesX, Y, Z and W respectively, and further calculating and correcting the X, Y, Z and W to generate a round key k₀，k₁，k₂And k₃And a modulation matrix I of size mxn;

the calculation and correction method of the 4 pseudo-random matrixes X, Y, Z and W is as follows:

where i 1,2, …, M, j 1,2, …, N, floor (t) denotes the largest integer returned which is less than or equal to t, mod (N, t) denotes the number N, the remainder to t.

Round key k₀，k₁，k₂And k is₃The calculation correction method is as follows:

the modulation matrix I calculation and correction method comprises the following steps:

I(i,j)＝mod((X(i,j)+Y(i,j)+Z(1,j)+W(i,j))*10⁵,255)+1 (5)

1.2 constructing Feistel structure, and carrying out bit-level scrambling, diffusion and pixel rearrangement operations on the image according to rows

A plaintext image P with the size of M multiplied by N is expanded into a one-dimensional vector R with the size of M multiplied by N in a line form, the vector R is grouped into a group of 4 pixels, the group is divided into (R/4) groups in total, and each group is { A ═ A_i,B_i,C_i,D_iI belongs to (1, T), wherein the number of pixels in the last group is less than or equal to 4, different Feistel structures are constructed according to the number of the pixels in the group, and then the vector C is generated after bidirectional bit level scrambling, diffusion and pixel rearrangement operations are carried out on the image;

for 4 pixels in a group, i.e. { A }_i,B_i,C_i,D_iAnd the bit-level scrambling and diffusing formula of the bidirectional operation Feistel structure is as follows:

after rearrangement the order of the pixels in the group is { D_1i,C_1i,B_1i,A_1i}；

For the last group of 3 pixels, i.e., { E, F, G }, the bit-level scrambling and diffusing formula for the bi-directional Feistel structure is:

after rearrangement the order of the pixels in the group is { G }₁,F₁,E₁}；

For the last group of 2 pixels, i.e., { H, I }, the bit-level scrambling and diffusing formula for the bi-directional operation Feistel structure is:

the order of the pixels in the group after rearrangement is { I }₁,H₁}；

For the last group there are 1 pixel, i.e. { J }, whose value is rewritten to (255-J).

1.3 constructing Feistel structure, and carrying out bit-level scrambling, diffusion and pixel rearrangement operations on the image according to columns

Resetting an image information matrix C1 according to the M multiplied by N mode of the vector C, then expanding the vector C into a one-dimensional vector C2 according to the form of columns, grouping the vector C2 also according to the form of a group of 4 pixels, wherein the last group is less than or equal to 4 pixels, and generating a one-dimensional vector C3 after carrying out bit-level scrambling, diffusion and pixel rearrangement on the vector C2 by using the method in 1.2;

2. global scrambling and diffusion of images

Carrying out global scrambling and diffusion operation on the image information vector C3 by using the modulation matrix I to generate a final ciphertext image matrix C4;

the modulation matrix I is expanded in the form of rows into an M × N sequence S (I), I ═ 1,2, …, M × N, according to the formula g (I) ═ mod (S) ((I))i)×10¹⁰M × N) +1 calculates the sequence g (i), leaves only the first bit of the repeated elements in g (i), adds the values of the set {1,2, …, M × N } that do not appear in g (i) to the end of g (i) in descending order, and scrambles them according to the following formula:

resetting the C3 to be an M multiplied by N image information matrix, and generating a final ciphertext matrix C4 after performing diffusion operation according to the following formula:

the hyper-chaos Lorenz system is combined with the Feistel technology, and the image encryption effect is achieved. The invention has the beneficial effects that: the Feistel structure is operated in two directions, so that the encryption efficiency of the image is improved; the initial value of the hyperchaotic system is generated by calculation and correction of a 256-bit hash value generated by a plaintext image by using an SHA-256 algorithm, so that on one hand, the key space of the algorithm is increased, on the other hand, the encryption process is closely associated with the plaintext image, and the capability of the algorithm for resisting plaintext attack is enhanced; the scrambling and diffusion operations are carried out on the plaintext image at both the bit level and the pixel level, so that the scrambling-diffusion effect is improved.

Drawings

FIG. 1 is a flow chart of an image encryption method based on chaotic mapping and bidirectional operation Feistel structure according to the present invention;

FIG. 2 is a schematic diagram of a bidirectional Feistel structure with 4 pixels in a group and a pixel rearrangement diagram according to the present invention;

FIG. 3 is a schematic diagram of a bidirectional Feistel structure and pixel rearrangement for 3 pixels in a group according to the present invention;

FIG. 4 is a schematic diagram of a bidirectional Feistel structure and pixel rearrangement for 2 pixels in a group according to the present invention;

FIG. 5 is an experimental result of 256 × 256Lena images in the present invention;

FIG. 6 is a histogram of a Lena plaintext image and a ciphertext image in accordance with the present invention;

FIG. 7 is a graph showing the correlation between adjacent pixels in a Lena plaintext image and a Lena ciphertext image according to the present invention;

Detailed Description

In order to further understand the technical solution of the present invention, the following further describes the embodiments of the present invention with reference to the accompanying drawings.

The invention discloses an image encryption method based on chaos mapping and bidirectional operation Feistel structure, which has a flow shown in figure 1 and comprises the following steps:

firstly, generating a 256-bit hash value by using an SHA-256 algorithm on plaintext image information, and calculating and correcting the hash value to generate an initial value x of the chaotic system₀，y₀，z₀And w₀And given parameter values, using the initial value iteration chaotic system to generate new values to update the initial values of the system.

And secondly, continuously iterating the chaotic system M multiplied by N times by using the updated initial value to generate 4 pseudo-random sequences, calculating and correcting the 4 pseudo-random sequences to generate 4 pseudo-random matrixes X, Y, Z and W respectively, further calculating and correcting the pseudo-random matrixes X, Y, Z and W, and generating a round key and a modulation matrix.

And thirdly, expanding the plaintext image into one-dimensional vectors in a line form, grouping the vectors into groups in a form of 4 pixels, and totally dividing the groups into T-R/4 groups, wherein each group is { A ═ A_i,B_i,C_i,D_iAnd e, i belongs to (1, T), the number of pixels in the last group is less than or equal to 4, different Feistel structures are constructed according to different numbers of pixels in the groups, and then bidirectional bit level scrambling, diffusion and pixel rearrangement operations are carried out on the image. The bidirectional Feistel structure and pixel rearrangement process for 4 pixels in a group is shown in fig. 2, the bidirectional Feistel structure and pixel rearrangement process for 3 pixels in a group is shown in fig. 3, and the bidirectional Feistel structure and pixel rearrangement process for 2 pixels in a group is shown in fig. 4.

And fourthly, expanding the operated image information into a one-dimensional vector according to a column form, and then carrying out bit-level scrambling, diffusion and pixel rearrangement operations according to the third step.

And fifthly, performing global scrambling and diffusion operation on the image information matrix by using the modulation matrix, and then generating a final ciphertext image matrix.

To verify the effectiveness of the present invention, the following simulation experiment further explains that the present invention is tested with Window 10(intel (r) core (tm) i5-4590,3.30GHZ, RAM 4.00GB) and Matlab 2017a as platforms, and the security of the encryption method of the present invention is analyzed from the perspective of resisting statistical and differential attacks. Fig. 5 is an experimental result of Lena images of 256 × 256. Where (a) in fig. 5 is a Lena plaintext image, (b) in fig. 5 is a Lena ciphertext image, and (c) in fig. 5 is a Lena decrypted image.

It can be seen from the figure that the ciphertext image subjected to the encryption method is similar to noise, any information about the plaintext image cannot be obtained from the ciphertext image, and the decrypted image is the same as the plaintext image, so that the aims of encrypting and decrypting the image are fulfilled.

1. Key space analysis

The security of the encryption method has a great relationship with the key space, and generally, the larger the key space is, the stronger the ability of the encryption method to resist brute force attack is. The key of the encryption method comprises a 256bit hash value and a round key k generated by an SHA-256 algorithm₀,k₁,k₂,k₃And an initial value x of the hyper-chaotic system₀,y₀,z₀,w₀. If the calculation accuracy is 10^-14Then the key space of the encryption method of the present invention is about 10¹⁵⁰Therefore, the encryption method has larger key space and can effectively resist violent attack.

2. Histogram analysis

The histogram represents the distribution frequency of image pixels and describes the statistical correlation of the image, and generally, the more uniform the histogram of the image pixel gray scale is, the more effectively the histogram can resist the attack of statistical analysis^[9]. Fig. 6(a) shows a gray level histogram of a plaintext image, and fig. 6(b) shows a gray level histogram of a ciphertext image after the encryption method of the present invention is used. FIG. 6 shows that the image encryption method of the present invention has good resistance to statistical analysisThe attacker cannot analyze the gray value distribution condition of the original image from the ciphertext image.

3. Pixel correlation analysis

The strong correlation exists among the pixels of the plaintext image, the correlation exists in the plaintext partial information, the information is easily utilized by lawless persons, and in order to resist statistical analysis, the correlation among the pixels needs to be reduced. The pixel correlation calculation formula is as follows:

where N is the logarithm of arbitrarily chosen neighboring pixels whose gray scale value is (u)_i,v_i) I 1,2, …, N, vector u { u ═ u_iV ═ v }, vector v ═ v_i}。

2000 pairs of adjacent pixel points are randomly selected from the plaintext image and the ciphertext image respectively, and correlation coefficients of the horizontal direction, the vertical direction and the diagonal direction are calculated, and the results are shown in table 1. The correlation diagrams of the adjacent pixels of the plaintext image in the horizontal direction, the vertical direction and the diagonal direction are shown in (a), (c) and (e) of fig. 7, and the correlation diagrams of the adjacent pixels of the ciphertext image in the horizontal direction, the vertical direction and the diagonal direction are shown in (b), (d) and (f) of fig. 7. From table 1 and fig. 7, it can be seen that the correlation coefficient between adjacent pixels of the plaintext image is close to 1, and the correlation coefficient between adjacent pixels of the ciphertext image is substantially 0, which indicates that the method of the present invention breaks the correlation between adjacent pixels, and a lawbreaker cannot effectively attack the adjacent pixels through statistical analysis.

TABLE 1 neighboring Pixel correlation comparison

4. Information entropy analysis

The information entropy reflects the uncertainty of the image, generally, the encryption effect of the algorithm is better, the information entropy of the image is closer to 8, and the information quantity and the randomness of the image are moreBig (a)^[9]. The calculation formula of the information entropy is as follows:

where L is the grey level of the image and p (i) represents the probability of the grey value i occurring.

The information entropy value of the plaintext image obtained through calculation is 7.4540, the information entropy value of the ciphertext image is 7.9917, the information entropy value of the ciphertext image is very close to the theoretical value 8, the information leakage possibility of the ciphertext is very low, and the method can further prove that the method can effectively resist statistical analysis attack.

5. Clear text sensitivity analysis

The plaintext sensitivity analysis means that two plaintext images with small difference are encrypted by using the same secret key through an encryption method to obtain two corresponding ciphertext images, the difference of the two ciphertext images is compared, if the two images have large difference, the encryption method has good plaintext sensitivity, and the plaintext sensitivity can be measured through a pixel change rate (NPCR) and a normalized average change rate (UACI), and the calculation formula is as follows:

where M, N represent the number of rows and columns of the image, respectively, c₁(i, j) represents the pixel value at position (i, j) of the original ciphertext image, c₂(i, j) represents the pixel value of the ciphertext image at position (i, j) after being slightly changed. If c is₁(i,j)＝c₂(i, j), D (i, j) is 0, otherwise D (i, j) is 1.

In order to detect the sensitivity of the invention to the plaintext, 1 is added to the pixel value of a certain position of the plaintext image, and then encryption is carried out. The values of NPCR and UACI of the ciphertext image before and after the pixel value of the plaintext image is changed are calculated as shown in table 2.

TABLE 2 values of ciphertext images NPCR and UACI before and after plaintext image pixel change

From table 2, it can be seen that the values of NPCR and UACI are both very close to their theoretical values, which indicates that the plaintext image changes slightly and the ciphertext image changes greatly, which indicates that the encryption method of the present invention is relatively sensitive to the plaintext, which further indicates that the encryption method of the present invention can effectively resist differential attack.

The initial value of the chaotic system is generated by plaintext information calculation, the sensitivity of the algorithm to the plaintext is enhanced, and the purposes of scrambling and diffusing the image bit level are achieved by constructing different Feistel structures and performing bidirectional encryption operation; the image information is subjected to global scrambling and diffusion by a pseudo-random sequence generated by the iterative hyper-chaotic system and a modulation matrix obtained through calculation and correction, so that the image information can effectively resist statistical attack and differential attack.

Reference to the literature

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[2]Ye Guodong,Pan Chen,Huang Xiaoling and Mei Qixiang.An efficient pixel-level chaotic image encryption algorithm[J].Nonlinear Dyn,2018,94(1):745-756.

[3]Chai Xiuli,Chen Yiran and Broyde Lucie.A novel chaos-based image encryption algorithm using DNA sequence operations[J].Opt Lasers Eng,2017,88:197-213.

[4]Ye Xiaolin,Wang Xingyuan,Gao Suo,et al.A new chaotic circuit with multiple memristors and its application in image encryption[J].Nonlinear Dyn,2019,99(2):1489-1506.

[5]Hua Zhongyun,Zhou Yicong and Huang Hejiao.Cosine-transform-based chaotic system for image encryption[J].Information Sciences,2019,480:403-419.

[6]Zhang Yong,Chen Aiguo,Tang Yingjun,et al.Plaintext-related image encryption algorithm based on perceptron-like network[J].Information Sciences,2020,526:180-202.

[7]Li Bo,Liao Xiaofeng and Jiang Yan.A novel image encryption scheme based on improved random number generator and its implementation[J].Nonlinear Dyn,2019,95(3):1781-1805.

[8]Liu Xu,Song Yurong and Jiang Guoping.Hierarchical bit-level image encryption based on chaotic map and Feistel network[J].International Journal of Bifurcation and Chaos,2019,29(2):1950016.

[9] Zhang Yong, chaos digital image encryption [ M ]. Beijing, Qinghua university Press, 2016:79-104.

Claims

1. An image encryption method based on chaos mapping and bidirectional operation Feistel structure is characterized by comprising the following steps:

firstly, generating a 256-bit hash value by using an SHA-256 algorithm on plaintext image information, and calculating and correcting the hash value to generate an initial value x of a hyperchaotic Lorenz system₀，y₀，z₀And w₀；

Using the SHA-256 algorithm, different plaintext images will generate different 256-bit hash values, represented in decimal notation as k₁,k₂,…,k₆₄Divide them into 4 groups, i.e. { k }₁,k₂,…,k₁₆}，{k₁₇,k₁₈,…,k₃₂}，{k₃₃,k₃₄,…,k₄₈And k₄₉,k₅₀,…,k₆₄Calculating an initial value x of the hyperchaotic Lorenz system according to the following formula₀，y₀，z₀And w₀After the initial value is used for iterating the hyper-chaos Lorenz system for 2000 times, the values of x, y, z and w are generated and serve as the updated initial value of the system;

secondly, continuously iterating the chaotic system M multiplied by N times by using the updated initial value to generate 4 pseudo-random sequences x (a)₁,a₂,…,a_M×N)，y(b₁,b₂,…,b_M×N)，z(c₁,c₂,…,c_M×N) And w (w)₁,w₂,…,w_M×N) Generating 4 pseudo-random matrixes X, Y, Z and W by calculating and correcting the 4 pseudo-random sequences, further calculating and correcting the pseudo-random matrixes, and generating a round key k₀，k₁，k₂And k₃And a modulation matrix I of size mxn;

where i 1,2, …, M, j 1,2, …, N, floor (t) denotes the largest integer returned which is less than or equal to t, mod (N, t) denotes the number N, the remainder to t;

I(i,j)＝mod((X(i,j)+Y(i,j)+Z(1,j)+W(i,j))*10⁵,255)+1 (4)

thirdly, expanding the plaintext image P with the size of M × N into M × N one-dimensional vectors R in a line form, grouping the vectors R into a group of 4 pixels, and dividing the group into (R/4) groups, each group being { a ═ b [ ] [ (] [ ]_i,B_i,C_i,D_iE, i belongs to (1, T), wherein the number of pixels in the last group is less than or equal to 4, different Feistel structures are constructed according to different numbers of pixels in the groups, and then the bidirectional bit level scrambling, diffusion and pixel rearrangement operations are carried out on the image to generate a vector C;

for 4 pixels in a group, i.e. { A }_i,B_i,C_i,D_i}, two-way operationThe bit-level scrambling and diffusing formula of the Feistel structure is as follows:

For the last group of 3 pixels, i.e., { E, F, G }, the bit-level scrambling and diffusing formula of the bidirectional operation Feistel structure is:

For the last group of 2 pixels, namely { H, I }, the bit-level scrambling and diffusing formula of the bidirectional operation Feistel structure is as follows:

the order of the pixels in the group after rearrangement is { I }₁,H₁}；

For the last group, 1 pixel, namely { J }, its value is rewritten to (255-J);

fourthly, resetting the image information matrix C1 according to the vector C by M multiplied by N, then expanding the vector C into a one-dimensional vector C2 in a column form, grouping the vector C into a group of 4 pixels, wherein the last group is less than or equal to 4 pixels, and generating a one-dimensional vector C3 after carrying out bit-level scrambling, diffusion and pixel rearrangement on the vector C2 by using the method of the third step;

fifthly, global scrambling and diffusion operations are carried out on the image information vector C3 by using the modulation matrix I, and then a final ciphertext image matrix C4 is generated;

the global scrambling process method is as follows:

the modulation matrix I is expanded in the form of rows into an M × N sequence s (I), I ═ 1,2, …, M × N, according to the formula g (I) ═ mod (s (I) × 10)¹⁰M × N) +1 calculates the sequence g (i), leaves only the first bit of the repeated elements in g (i), adds the values of the set {1,2, …, M × N } that do not appear in g (i) to the end of g (i) in descending order, and scrambles them according to the following formula:

the global diffusion process method is as follows:

c3 is reset to an M × N image information matrix, and a final ciphertext matrix C4 is generated after performing a diffusion operation according to the following equation.